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Subsection 7.1.1 Introduction

When we try to translate English sentences into logical statements, we may get stuck as we try to figure out exactly what the English sentences actually mean.

We could write volumes on the subject of English semantics: the process by which meaning is assigned to utterances. We won’t do that here. What we will do is just to sketch a few issues so that we’re aware, as we use English to describe the reasoning that we’ll do, of possible misunderstandings or confusion.

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And, by the way, English isn’t special in this regard. All natural human languages (Spanish, Urdu, Khmer, or whatever) have the problems that we are describing here, although the details differ.

Big Idea

English sentences can be ambiguous, vague, unclear, and sometimes even downright misleading.

The language of logic helps us avoid those problems.

But, to use it to express our ideas, we need to understand how it relates to our natural language, English.

Exercises Exercises

1.

We might think that it should be very straightforward to translate into logic claims about particular named individuals. For example, assuming that it is clear which person named Logan we are talking about, we can write:

Student(Logan)

But even this is not always so simple. Suppose that we want to encode into logic the sentence:

There is a tooth fairy.

Which one or more of the following expressions is a syntactically legal predicate logic statement (or wff), that may reasonably encode our claim if we have defined all the predicates appropriately:

I. ∃x (ToothFairy(x))

II. ∃ (ToothFairy)

III. ∃ Tooth Fairy

IV. ∃x (BringsMoneyForTeeth(x))

  1. Just I.

  2. Just II.

  3. Just III.

  4. Just IV.

  5. Just I and II.

  6. Just III and IV.

  7. At least three of them.

Answer.
Correct answer is F.
Solution.
Explanation: I and II are not syntactically correct. If we introduce a quantifier, we can build a legal wff only with one of these two forms: x (P) or x (P), where P is a wff. Both III and IV are legal wffs. Whether or not they say what we intend depends completely on the definitions of the predicates that they use.

2.

Suppose that we want to encode into logic the sentence:

Santa Claus does not exist.

Which one or more of the following expressions is a syntactically legal predicate logic statement (or wff), that may reasonably encode our claim if we have defined all the predicates appropriately

I. ¬∃ (SantaClaus)

II. ¬∃ SantaClaus

III. ¬∃x (SantaClaus(x))

IV. ¬∃x (WearsRedSuit(x) ∧ LivesAtNorthPole(x) ∧ ClimbsDownChimneys(x) ∧ BringsXmasPresents(x))

Answer.
Solution.

Big Idea

In many cases, if we try to map from English sentences to logical ones without clearly analyzing what the English sentence is saying, we will have trouble. It may be tempting to jump to the conclusion that we have found a weakness in our logical system itself. While there are real limits to the logical system that we are discussing, we should, before assuming that we’ve found one, see if a more careful translation of English into logic can’t make the problem disappear.